4 Answers
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"Groove" has to do with taking a strict metronomic beat and then striking certain notes in each measure either slightly ahead of, or slightly behind, the strict metronomic beat. In a rock music groove, for instance, the dominant beats in 4/4 time are beats 2 and 4. The notes on those beats would be sounded in strict, unwavering time. The "groove" would be created by varying other notes in each measure in a subtle fashion.

Which notes to vary in time, and by what degree, is what creates different kinds of grooves.

The term "groove" first came into play in mid-to-late African-American styles such as R&B ("Rhythm and Blues") and later funk.

The precursor terms to "groove" are "swing" and "shuffle", which describe specific kinds of shifting of rhythmic emphasis that first appeared in jazz, which again, is ultimately an African-American musical innovation. Swing and shuffle have to do with taking a 4/4 rhythm and imposing an underlying 12/8 rhythm with shifting in emphasis of the eighth-note offbeats of 4/4 so that they almost take on a triplet feel, shifting that off-beat eighth note to line up with the third eighth note in a 12/8 grouping. There is much written on this subject. For an understanding of "swing", study dance band jazz from the 1930s or 1940s, such as that by Count Basie, Duke Ellington, or Glenn Miller. For a good example of "shuffle", listen to Texas-style blues-rock like ZZ Top.

I must insist that you will never understand or appreciate "groove" if you only listen to music that is not from the African-American tradition and is from the last thirty years or so. Start with African-American music from the USA going back at least to the mid-20th century.

This answer is by necessity very technical, describing what groove is, but I think the points made here are correct: you really want to listen to and learn a wide range of music types to learn the feel of a groove, because it does all come from feeling and physical movement.
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Dr Mayhem♦Aug 25 '11 at 13:26

Precursors -- okay. So does the "groove" means exactly the same as "shuffle" or is it differs in the amount of shift?
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horshNov 24 '11 at 8:58

There are many general applications of the term "groove". "Shuffle", or "swing" on the other hand is just one example of creating a groove. "Shuffle" is a the specific case of making a 4/4 meter sound more like 12/8.
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Wheat WilliamsJan 19 '12 at 18:59

Great answer although I think th term is now used more broadly (rightly or wrongly) to mean a great "rhythmic feel". As some of this is pretty straightforward without the shuffle feel etc, it's kind of bent the meaning of the word.
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user2808054Oct 20 '14 at 16:29

I would recommend reading 'The Music Lesson' by bassist, Victor Wooten. He describes groove as the most important element in music. (Examples of some of the other elements are notes, space, tone, rhythm, dynamics, etc.)

I am a student of his Music & Nature camp outside of Nashville, TN, and I feel he does an amazing job of teaching this concept. He is able to articulate what it is that makes groove so important.

Victor teaches that of the 10 main elements which make up music, groove is the most important. The best lesson I've ever experienced (in my 20+ years as a pianist) was a session with him showing that even if you do everything else (#2 through #10) RIGHT, -without groove, your performance wont be as good. Most of us spend most of our musical careers focused on elements 2-10. Victor shows again and again how, even a musician who may not have 2-10 mastered will sound like an awesome player if they understand and know how to 'groove'.

I would say that groove is the way in which a musician emphasizes rhythms and notes to create a 'feel'. It's that special ingredient in performance that brings a musical sequence to life. For the musician/s performing it, as well as the listener.

Groove is the sense of propulsive rhythmic "feel" or sense of "swing" created by the interaction of the music played by a band's rhythm section (drums, electric bass or double bass, guitar, and keyboards). Groove is a consideration in genres such as salsa, funk, rock, fusion, and soul. The word is often used to describe the aspect of certain music that makes one want to move, dance, or "groove".

Groove is technically achievable, otherwise it wouldn't exist. Our friend above defined this very well as superimposing a triplet feel over 4/4. The people with amazaing techical levels who can't groove are like their beats-square.
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user12178Jun 18 '14 at 17:08

I think the groove mainly is in half-time feel stuff, you know, like breakdowns, try listening to Pantera's Revolution Is My Name, and you'll feel the groove when a certain part of the song comes up.
Also I think that bass lines are a major contributor in grooves.